Abstract: | Let $\{u(t, x)\}_{(t,x)\in\mathbb{R}_+\times \mathbb{R}}$ be the density of one-dimensional super-Brownian motion starting from Lebesgue measure. Using the Laplace functional of super-Brownian motion, we prove that as $N\to\infty$, the normalized spatial integral $N^{-1/2}\int_0^{Nx}[u(t, z)-1]\mathrm{d}z$ converges jointly in $(t, x)$ to Brownian sheet in distribution. This is based on joint work with Zenghu Li. |